V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
VLADIMIR BENIĆ
SONJA GORJANC
Review
Accepted 13.12.2006.
(1,n) Congruences
(1,n) Congruences
(1,n) kongruencije
ABSTRACT
SAŽETAK
The first order algebraic congruences are classified into
two basic classes which depend on their directing curves.
By the method of synthetic geometry, we investigated the
basic properties for each of these classes: the construction
of rays, singularities, decomposition into developable surfaces, focal properties and the types of rays. The paper
ends with a short analytical approach, which enables the
visualizations of these congruences in the program Mathematica. Some exmples are shown.
Algebarske kongruencija prvoga reda razvrstane su u dvije
osnovne klase, ovisno o njihovim ravnalicama. Za svaku od
tih klasa, metodologijom sintetičke geometrije, istražuju se
osnovna svjstva: konstrukcija zraka, singulariteti, dekompozicija na razvojne plohe, žarišne osobine te vrste zraka.
Na kraju se daje i kratki analitički pristup koji je omogućio
izradu programa za vizualizaciju ovih kongruencija u programu Mathematica. Pokazano je nekoliko primjera.
Key words: congruence, decomposition on developable
surfaces, focal lines, singularities, visualization
Ključne riječi: kongruencija, dekompozicija na razvojne
plohe, žarišna linija, singulariteti, vizualizacija
MSC 2000: 51M15,51M30,51N10,51N35,68U05
1 Introduction
A congruence C is a double infinite line system, i.e. it is a
set of lines in a three-dimensional space (projective, affine
or Euclidean) depending on two parameters. A line l ∈ C
is said to be a ray of the congruence.
The order of a congruence is the number of its rays which
pass through an arbitraty point; the class of a congruence
is the number of its rays which lie in an arbitrary plane.
mth order, nth class congruence is signed C nm .
A point is called the singular point of a congruence if ∞1
rays pass through it. A plane is called the singular plane
of a congruence if it contains ∞1 rays.
Rays in a congruence can be decomposed in two ways into
a one-parameter family of developable surfaces (torses) so
that through every ray l ∈ C pass two torses that are real
and different (the case of hyperbolic ray), or imaginary (an
elliptic ray), or real and coincident (a parabilic ray).
The points of contact of a ray l ∈ C with the edges of regression (cuspidal edges) of these torses are called the foci
of l. The foci of l are the intersection points of l with consecutive rays of a congruence. The surfaces formed by the
foci of the rays of a congruence are called its focal surfaces. Each ray of a congruence touches its focal surface
at the foci. Two planes defined as the planes containing a
ray and consecutive rays are the focal planes of a congruence. The focal surface is the envelope of the focal planes.
A congruence clearly reciprocates into a congruence. The
focal planes and points are interchanged and the focal surface reciprocates into the new focal surface.
[11], [5], [9], [1]
Since the lines of the congruence are bitangents of the focal surface, every congruence of lines may be regarded as
the system of bitangents of a surface. The surface may,
however, break up into two separate surfaces, and the original surface, or each or either of the component surfaces
may degenerate into a curve; we have thus as congruences
the following systems of lines:
1. the bitangents of a surface,
2. common tangents of two surfaces,
3. tangents to a surface from the points of a curve,
4. common transversales of two curves,
5. lines “through two points” of a curve,
where the last four cases being degenerate cases of the first,
which is the general one. [9, p.37]
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V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
2 1st order congruences
It was proved that the rays of the first order congruences
are always tranversales of two curves, or they intersect the
same space curve twice. Beside that it was proved that the
only congruence of the first order, consisting of a system
of lines meeting a proper curve twice, is when the curve is
a twisted cubic. ([9, p. 64], [14, pp. 1184-1185], [13, p.
32])
The nth order space curve can possess singular points with
the highest multiplicity n − 2. If the directing curve cn has
a multiple point it must lie on the line d because if cn had a
mutiple point out of the line d, the plane through that point
and the line d would cut the curve cn in more than n points,
which is impossible. The k-ple point of the curve cn which
lies on the line d is signed Dki , where i ≤ n − k. Three examples of 7th and 8th order curves with dobule and triple
points are shown in Fig. 2.
If a congruence C is a system of lines meeting two directing curves of the orders m and m, which have α common
points, the order of a congruence is mm, − α. The only
congruence of the first order of this kind is when the directing curves are a curve of the nth order and a straight
line meeting it n − 1 times. [9, p. 64]
Therefore, we have only two types of the congruences of
the first order:
Type I 1st odrer nth class congruences C n1 are the systems
of lines which intersect a space curve cn of the order
n and a straight line d, where cn and d have n − 1
common points.
Type II 1st order 3rd class congruence B 31 is the system of
lines which meet a twisted cubic twice.
2.1 Congruences of the type I
2.1.1 Directing lines of C n1
The directing lines of a congruence C n1 are a space curve
cn of the order n and a straight line d which intersects cn
in n − 1 points. If all intersection points are the regular
points of cn we will sign them D11 , ....., D1n−1 . Some of
these points can coincide. There are cases when the line
d is the tangent line of cn , the tangent at inflection, etc. If
cn and d have s-ple contact at one regular common point it
is signed Di1,s , where i ≤ n − s. (See Fig. 1)
Figure 2
2.1.2 Singular points of C n1
a) All singular points of C n1 (the points which contain ∞1
rays of C n1 ) lie on its directing lines cn and d.
j
If a point C lies on the curve cn and C 6= Di , then the rays
of Cn1 which pass trough C form the pencil of lines (C) in
the plane δ ∈ [d] which contains C and d. (See Fig. 3)
Figure 3
j
Figure 1
6
b) If a point D lies on the line d and D 6= Di , then all the
lines which join D with the points of the curve cn are the
rays of C n1 . They form nth degree cone ζnD with the vertex
D. Since cn and d have n − 1 common points, this cone
intersects (or tuoches) itself n − 1 times through the line d,
thus the line d is (n − 1)-ple generatrix of ζnD . (See Fig. 4)
V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
2.1.3 Rays of C n1 through an arbitrary point
Every point A which is not the singular point of C n1 , i.e.
A∈
/ cn , A ∈
/ d, determines the plane δA ∈ [d] which cuts cn
in only one point C which in general does not lie on the line
d. The line AC, which cut d in one point D, is the unique
ray of C n1 through the point A. If the plane δA contains one
of the the tangent lines of cn at the intersection point Dki (or
if it is the rectifying plane at Di1,s ), then the points C and
D cioncide with Dki (Di1,s ) and the line ADki (ADi1,s ) is the
unique ray of C n1 through A. (See Fig. 6)
Figure 4
c) If a point Dki is the intersection point of cn and d and
if it is a k-ple point of the curve cn , then the rays through
Dki which cut cn form (n − k)th degree cone ζn−k
with the
Dk
i
vertex Dki . The line d is (n − k − 1)-ple genetartix of ζn−k
.
Dk
i
Besides that the rays through the point Dki form k pencils
of lines (Dki ) in the planes determined by the line d and k
tangent lines of cn at Dki . If the intersection point is Di1,s ,
then the pencil of lines (Di1,s ) in the rectifying plane of cn
at Di1,s are also the rays of a congruence.
The other lines of the sheefs {Dki } and {Di1,s } are not regarded as the rays of a congruence.
The example of the rays through the regular intersection
point D11 is shown in Fig. 5.
Figure 6
2.1.4 Singular planes of C n1
All singular planes of C n1 (the planes which contain ∞1 rays
of C n1 ) are the planes of the pencil [d]. From 2.1.2. it is
clear that in every plane δ ∈ [d] lie thes pencil of rays (C)
or (Dki ) or (Di1,s ). (See Fig. 7)
It is possible that some of the tangent lines at intersection
points Dkj lie in the same plane of the pencil [d]. In such
case there is more than one pencil of lines in the plane determined by these coplanar tangent lines.
Figure 7
Figure 5
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V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
2.1.5 Rays of C n1 through an arbitrary plane
Every plane α which is not the singular plane of C n1 , i.e.
α∈
/ [d], contains n rays of the congruence. The plane α
cuts line d in one point D and nth order space curve cn in
n points C j , j = 1, ..., n. The lines DC j are n rays of the
congruence C n1 in the plane α. They are the intersection of
the plane α and nth degree cone ζnD and can be real and different, coinciding or imaginary. If α cuts the line d in Dki ,
then n − k rays are the intersection of α and the cone ζn−k
Dk
i
and other k rays are the intersection of α and the planes
through d determined by the tangent lines of cn at Dki . (See
Fig 8)
and the foci are imaginary. If the developables coincide,
the ray is parabolic and the foci coincide. A congruence
or the partition thereof is said to be hyperbolic, elliptic or
parabolic if its rays are hyperbolic, elliptic or parabolic.
In the case of the 1st order surfaces of the type I the focal surface degenerates into the directing curves cn and d.
The developables have not the cuspidal edges, only cuspidal points: the verteces D of the cones ζnD and the points
C which are the intersections of the planes δ ∈ [d] and the
curve cn . Thus, each ray of C n1 has the foci on the directing
lines cn and d. If they are real, the congruence is hyperbolic
with parabolic rays in n − 1 planes through the points Dki
where the developables and foci coincide. If the directing
lines are imaginary the congruence is elliptic.
2.2 Congruences of the type II
If a congruence B is the set of lines which cut a proper
curve twice, this curve must be a twisted cubic. Since
through an arbitrary point only one ray of the 1st order
congruence passes, the projection of the directing curve
from this point onto an arbitrary plane has only one double
point. Thus this projection is the 3rd order plane curve. As
the original curve and its projection have the same order,
then the directing line of a congruence B is a twisted cubic.
The projection of a twisted cubic onto a plane from a point
on a secant line yields a nodal cubic and from a point on a
tangent line a cuspidal cubic [4, p. 54]. (See Fig. 9)
Figure 8
2.1.6 Decomposition of C n1 into developable surfaces
As mentioned in the introduction every congruence can be
decomposed in two ways into a one-parameter family of
developables. These two families arise if one of the two
parameters of which a congruence depends, is fixed.
The tangent and a secant lines of a twisted cubic b3 fill
up the projective space and are pairwise disjoint, except at
points at curve itself [4, p. 90]. Thus through an arbitrary
point unique ray of the congruence B passes.
In the case of the 1st order congruences of the type I the
devlopables are the sets of rays through singular points, i.e.
one family is formed by the nth degree cones ζnD , and the
other by the planes of the pencil [d]. Every ray of C n1 is
the intersection of two developables, one from each family. Since d is (n − 1)-ple generatrix of ζnD , every plane of
[d] cuts it into only one more generatrix which is the ray of
a congruence. For the rays through the intercestion points
Dki the nth degree cones split into (n − k)th degree cone and
k planes through the line d. (See Figures 4, 5 and 7)
2.1.7 Focal properties − hyperbolic,
parabolic rays of C n1
elliptic and
In general case the ray of a congruence touches the focal
surface at foci which lie on the cuspidal edges of the developables. If the developables through the ray are real and
different, the ray is hyperbolic and the foci are real and distinct. If the developables are imaginary, the ray is elliptic
8
Figure 9
V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
It is clear that such congruence is of the 3rd class (B 31 ),
because every plane cuts the curve b3 in 3 points and the
lines joining them are three rays of a congruence. These
three rays can be: three real and different (a), one real and
two imaginary (b), three real where two of them coincide
(c) and three real and coinciding (d). (See Fig. 10)
of B 31 which cuts b3 at the points B1 , B2 is the part of the
intersecion of the cones ζ2B1 , ζ2B2 . Namely, the intersection
of ζ2B1 and ζ2B2 is b3 ∪ B1 B2 .
The curve b3 is the focal curve of B 31 , it contains the cuspidal points of ζ3B , B ∈ b3 . The ray of B 31 is hyperbolic if
the intersection points B1 , B2 ∈ b3 are real and different, it
is parabolic if they coincide (the ray is a tangent of b3 ) and
it is elliptic if they are imaginary.
The parabolic rays of B 31 form the tangent developable of
b3 . (See Fig. 12)
Figure 12
Figure 10
All singular points of B 31 lie on the twisted cubic b3 . The
lines which join the point B ∈ b3 with the other points of
b3 form 2nd degree cone ζ2B . (See Fig. 11)
The rays of such congruences are also the intersections
of the corresponding elements of two collinear bundles of
planes {B1 }, {B2 }, [7, p. 135], [14, p. 1185]. In this case
the basic points of the bundles (B1 , B2 ) lie on a twisted cubic b3 , and the unique ray through an arbitrary point can
be constructed as the part of the intersection of two ruled
quadrics. These quadrics pass through b3 and their rulings are determined by the collineation between the bundles {B1 } and {B2 }, [7, p. 136]. In the special case when
one plane in the collineation between {B1 } and {B2 } corresponds to itself, the basic cubic b3 splits into one straight
line and a conic which have one common point, and the
congruence B 31 splits into the 2nd class congruence C 21 and
the field of lines in the plane of the conic. These congruences are elaborated in detail in [2].
3 Analytical approach and Mathematica
visualizations
If two algebraic space curves c1 and c2 are given by the
following parametric equations
Figure 11
c1 ... x = x1 (u),
Since every plane contains exactly three rays of B 31 there
are no singular planes of B 31 .
B 31 can be decomposed into one family of developables.
This family consists of the cones ζ2B , B ∈ b3 . Every ray
y = y1 (u),
z = z1 (u),
x1 , y1 , z1 : I1 → R, I1 ⊆ R, x1 , y1 , z1 ∈ C1 (I1 )
c2 ... x = x2 (v),
y = y2 (v),
z = z2 (v),
x2 , y2 , z2 : I2 → R, I2 ⊆ R, x2 , y2 , z2 ∈ C1 (I2 ),
(1)
9
V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
then the set of lines which join the points of c1 and c2 are
given by the following equations
y − y1 (u)
z − z1 (u)
x − x1(u)
=
=
,
x1 (u) − x2 (v) y1 (u) − y2(v) z1 (u) − z2 (v)
(2)
(u, v) ∈ I1 × I2 ⊆ R2 .
In the previous section, for drawing the directing lines of
C n1 , we used the following parametric functions:
d ... x1 (u) = 0, y1 (u) = 0, z1 (u) = u, u ∈ R,
cn ... x2 (v) = ax (v − v1) · · · (v − vn−1),
y2 (v) = ay v x2 (v),
z2 (v) = v,
v, ax , ay , v1 , . . . , vn−1 ∈ R.
(3)
It is clear that the line d is the axis z and cn is the nth order
space curve which cuts the axis z at the points Di (0, 0, vi ),
i ∈ {1, ..., n − 1}.
If the polynomial x2 (v), from (3), contains the factor
(v − vi )s , then i ≤ n − s and d and cn have s−ple contact
at the point Di (0, 0, vi ).
If the polynomial z2 (v), from (3), takes the form
z2 (v) = v(v − vi1 ) · · · (v − vik ), vi j 6= 0,
i1 , . . . , ik ∈ {1, . . . , n − 1},
(4)
k ≤ n − 2,
then (0, 0, 0) is the k−ple singular point of cn and the coordinates of intersection points of cn and d are (0, 0, z2 (vi )).
(3) and (2) give the following equations of the rays of C n1
y
u−z
x
=
=
,
x2 (v) y2 (v) u − z2 (v)
(u, v) ∈ R2 .
(5)
The above equations enable computer visualization of the
rays of C n1 and B 31 . Based on this we made the program in
webMathematica which enables interactive visualizations
of C n1 on the internet. It is available at the following address:
www.grad.hr/itproject math/Links/webmath/indexeng.html
Figure 13
E XAMPLE 2
The rays of the 4th class congruence are shown in Fig. 14.
The directing lines of this congruence are the Viviani’s
curve c4 which cuts the axis z in two points, but one of
the intersection points is the double point of c4 . The parametric equations of c4 are:
√
2
v
x(v) =
(cos v − 2 sin − 1), y(v) = sin v,
2
√2
2
v
(cos v + 2 sin − 1), v ∈ [−2π, 2π].
z(v) =
2
2
3.1 Examples
In the following examples the graphics are produced with
the program Mathematica.
E XAMPLE 1
Two dsiplays of the same 2nd class congruence are shown
in Fig. 13. The directing lines of this C 21 are the axis z and
the circle given by the following parametric equations:
x(v) = cosv + 1, y(v) = sin v, z(v) = 0, v ∈ [0, 2π].
10
Figure 14
V. Benić, S. Gorjanc: (1,n) Congruences
KoG•10–2006
E XAMPLE 3
E XAMPLE 4
Two displays of the same 7th class congrence are shown in
Fig. 15. The directing lines of this C 71 are the axis z and
the curve c7 which is given by the following parametric
equations:
The visualization of B 31 whose directing curve is given by
the following parametric equations
1
v(v − 1)(v − 2)(v − 3)(v − 3.5)(v − 4),
10
y(v) = 2 v x2 (v),
x(v) =
z(v) = v,
v ∈ R.
x(v) = v,
y(v) = (v − 1)(v + 1),
z(v) = (v − 1)2(v + 1),
v∈R
is shown in Fig. 16. The same congruence, with red
parabolic rays, is shown in Fig. 17 for two different view
points.
Figure 16
Figure 15
Figure 17
11
KoG•10–2006
V. Benić, S. Gorjanc: (1,n) Congruences
References
[9] G. S ALMON : A Treatise on the Analytic Geometry of
Three Dimensions, Vol.II . Chelsea Publishing Company, New York, 1965. (reprint)
[1] S. P. F INIKOV:
Theorie der Kongruenzen.
Akademie-Verlag-Berlin, Berlin, 1959. (translation from Russian)
[2] S. G ORJANC : The Classification of the Pedal Surfaces of (1,2) Congruences. Dissertation, Department of Mathematics, University of Zagreb, 2000. (in
Croatian)
[3] A. G RAY: Modern Differential Geometry of Curves
and Surfaces with Mathematica. CRC Press, Boca
Raton, 1998.
[4] J. H ARRIS : Algebraic Geometry: A First Course
Springer, New York, 1992.
[5] C. M. J ESSOP : A Treatise on the Line Geometry. Chelsea Publishing Company, New York 1969
(reprint).
[6] E. M ÜLLER , J. L. K RAMES : Konstruktive Behandlung der Regelflächen. Franc Deuticke, Leipzig und
Wien, 1931.
[7] V. N I ČE : Introduction to Synthetic Geometry.
Školska knjiga, Zagreb, 1956. (in Croatian)
[8] G. S ALMON : A Treatise on the Analytic Geometry of
Three Dimensions, Vol.I . Chelsea Publishing Company, New York, 1958. (reprint)
12
[10] G. S ALMON : Higher Plane Curves. Chelsea Publishing Company, New York, 1960. (reprint)
[11] Encyclopaedia of Mathematics - SpringerLink.
Edited by M. Hazewinkel, available online: http:
//eom.springer.de/C/c024910.htm
[12] R. S TURM : Die Lehre von den geometrischen Verwandtschaften, Band IV. B. G. Taubner, LeipzigBerlin, 1909.
[13] R. S TURM : Liniengeometrie, II. Teil. B. G. Taubner,
Leipzig, 1893.
[14] K. Z INDLER : Algebraische Liniengeometrie. Encyklopädie der Mathematischen Wissenschaften, Band
III, 2. Teil, 2. Hälfte. A., pp. 1184-1185, B. G. Teubner, Liepzig, 1921-1928.
Vladimir Benić
e-mail: [email protected]
Sonja Gorjanc
e-mail: [email protected]
Faculty of Civil Engineering University of Zagreb
Kačićeva 26, 10000 Zagreb